An Introduction to Stochastic Modeling

An Introduction to Stochastic Modeling provides information pertinent to the standard concepts and methods of stochastic modeling. This book presents the rich diversity of applications of stochastic processes in the sciences.

Organized into nine chapters, this book begins with an overview of diverse types of stochastic models, which predicts a set of possible outcomes weighed by their likelihoods or probabilities. This text then provides exercises in the applications of simple stochastic analysis to appropriate problems. Other chapters consider the study of general functions of independent, identically distributed, nonnegative random variables representing the successive intervals between renewals. This book discusses as well the numerous examples of Markov branching processes that arise naturally in various scientific disciplines. The final chapter deals with queueing models, which aid the design process by predicting system performance.

This book is a valuable resource for students of engineering and management science. Engineers will also find this book useful.

Preface

Chapter I Introduction

1.1 Stochastic Modeling

1.2 Probability Review

1.3 The Major Discrete Distributions

1.4 Important Continuous Distributions

1.5 Some Elementary Exercises

1.6 Useful Functions, Integrals, and Sums

Chapter 2 Conditional Probability and Conditional Expectation

2.1 The Discrete Case

2.2 The Dice Game Craps

2.3 Random Sums

2.4 Conditioning on a Continuous Random Variable

Chapter 3 Markov Chains: Introduction

3.1 Definitions

3.2 Transition Probability Matrices of a Markov Chain

3.3 Some Markov Chain Models

3.4 First Step Analysis

3.5 Some Special Markov Chains

3.6 Functionals of Random Walks and Success Runs

3.7 Another Look at First Step Analysis

Chapter 4 The Long Run Behavior of Markov Chains

4.1 Regular Transition Probability Matrices

4.2 Examples

4.3 The Classification of States

4.4 The Basic Limit Theorem of Markov Chains

4.5 Reducible Markov Chains

4.6 Sequential Decisions and Markov Chains

Chapter 5 Poisson Processes

5.1 The Poisson Distribution and the Poisson Process

5.2 The Law of Rare Events

5.3 Distributions Associated with the Poisson Process

5.4 The Uniform Distribution and Poisson Processes

5.5 Spatial Poisson Processes

5.6 Compound and Marked Poisson Processes

Chapter 6 Continuous Time Markov Chains

6.1 Pure Birth Processes

6.2 Pure Death Processes

6.3 Birth and Death Processes

6.4 The Limiting Behavior of Birth and Death Processes

6.5 Birth and Death Processes with Absorbing States

6.6 Finite State Continuous Time Markov Chains

6.7 Set Valued Processes

Chapter 7 Renewal Phenomena

7.1 Definition of a Renewal Process and Related Concepts

7.2 Some Examples of Renewal Processes

7.3 The Poisson Process Viewed as a Renewal Process

7.4 The Asymptotic Behavior of Renewal Processes

7.5 Generalizations and Variations on Renewal Processes

7.6 Discrete Renewal Theory

Chapter 8 Branching Processes and Population Growth

8.1 Branching Processes

8.2 Branching Processes and Generating Functions

8.3 Geometrically Distributed Offspring

8.4 Variations on Branching Processes

8.5 Some Stochastic Models of Plasmid Reproduction and Plasmid Copy Number Partition

8.6 Population Growth Processes with Interacting Types

8.7 Deterministic Population Growth with Age Distribution

Chapter 9 Queueing Systems

9.1 Queueing Processes

9.2 Poisson Arrivals and Exponentially Distributed Service Times

9.3 The M/G/1 and M/G/∞ Systems

9.4 Variations and Extensions

9.5 Open Acyclic Queueing Networks

9.6 General Open Networks

Further Readings

Solutions to Selected Problems

Index